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d d HOLE NO. 162 
1d eit 7 A Ae REVIEW PUBLICATIONS ™ eer 


Psychological Monographs 


EDITED BY 


SHEPHERD IVORY FRANZ, Univ. or Catir., So. BRANCH 
HOWARD C. WARREN, Princeton University (Review) 
JOHN B. WATSON, New York City (Review) 
MADISON BENTLEY, University or Itirnois (J. of Exp. Psych.) 
S. W. FERNBERGER, UNIversity ofr PENNSYLVANIA (Bulletin) and 
W. S. HUNTER, Crark University (Index) 


On the Melodic Relativity of Tones 


BY 
OTTO ORTMANN 


PEABODY CONSERVATORY OF MUSIC 
BALTIMORE, MD. 


PSYCHOLOGICAL REVIEW COMPANY 


PRINCETON, N. J. 
AND ALBANY, N. Y. 


Acents: G. E. STECHERT & CO., Lonpvon (2 Star Yard, Carey St., W. C.) 
Lerezic (Hospital St., 10); Paris (76, rue de Rennes) 


3 


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ras 4 ay) 4 i 
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Mak ‘ 


lesa 4 sey & 





CONTENTS 


PAGE 

ERE TROT N ree RP Redes cK eV Ean en ar ueNe ICO Une 1 
TP VERTIC ASPECTGE MELODYiGe ear ed see aU al 4 
ALES OLUEI CUE ICLOT Ss PAC CE OR SS em BT EAN ek 4 

UR CULLLETCR TAGE OT SEM Ne DS EY IE GRP IED OY 5 7 

Rigg EMPORALAASPECTS OF, MELODY...) Gi ae 19 
Ill. INFLUENCE oF EXTRANEOUS FACTORS.............. 26 
IV. Errects or Metopic RELATIONSHIP............... 34 
V. A TEST IN MELODIC MEMORY........... ria at ate ‘ 38 
BDOMCEUSIONG Nine cut en tue LUNE Tele ramon Muar ana mt GeD 46 


BORDER RNC BS ial ean eMi UN WAL UNA de |LOSS ANM CPi )N lib Sine si 1c 47 


Digitized by the Internet Archive 
In 2022 with funding from 
Princeton Theological Seminary Library 


httos://archive.org/details/onmelodicrelativOOortm 


INTRODUCTION 


Melody, in its broadest connotation, means any succession of 
single tones. Such a definition includes many examples of tonal 
combinations not considered melody by the musician, layman, or 
by the psychologist. The inclusion is necessary, however, in 
order to explain certain reactions to tones. 

The definition usually given (we find it in various forms in the 
works of Stumpf, Mach, Bingham, and others) demands the 
existence of some finality, cadence, law of return, or acceptance 
of the melody by peoples at large. To some extent these attri- 
butes are the result of the operation of more basic psychological 
laws, and to a further extent they involve the operation of other 
musical, but non-melodic elements such as harmony, tonality, and 
rhythm. So long as we treat melody in connection with harmony 
and rhythm we cannot isolate any purely melodic factors. The 
striking effect of these attributes may be seen if we change the 
rhythm of familiar melodies. The following example, Fig. 1, 
of which only the rhythm has been changed in a simple manner 
(it remains in uniform meter [4/4] throughout), will be recog- 
nized with some difficulty as “America,” and its acceptance as 
a desirable melody, in the form here given, is doubtful. The 
effect of harmony is no less pronounced. The following tonal 
sequences, Fig. 2, seem to be lacking in finality, coherence and 
musical value. Yet if they be harmonized, let us say as in Fig. 3, 
these attributes and some esthetic satisfaction appear. Similar, 
though perhaps less pronounced, effects are present in all artistic 
and folk-tune melody, and unless we exclude these attributes, we 
cannot use either the art-song or the folk-song as a basis for an 
analysis of the purely melodic relationship of tones. 

Melody is motion in the pitch-series. This motion results 
from a succession of pitch distances or interval, in which the 
repeated tone counts as a zero interval. The problem here is not 
to seek the causes of the acceptance of some and the rejection of 
other forms of pitch-motion as desirable, but to seek the melodic 
relationships existing among the tones of any melody, regardless 


OTTO ORTMANN 





ON THE MELODIC RELATIVITY OF TONES 3 


of its esthetic or form value. Melodic-relativity thus becomes 
the equivalent of pitch-relativity, and our problem is not the con- 
sideration of the melody as a whole, a musical or artistic unit, 
but of the pitch-relationship of each tone to all the other tones 
of the tonal sequence selected. 

Since melody is pitch-motion, the attributes of the pitch-series 
will form the basis of the investigation. Chief among these is 
continuity. The pitch-series is a continuous stretch of sensory 
material, unbroken by any periodicity. Between C and Ct, for 
example, lies a short stretch of this series, which may contain 
any number of intermediate pitch-points. This is best illustrated 
by instruments such as the siren, the sliding intonation of which 
eliminates all pitch-points. Any “slide” thus produced (the 
natural example of which is the howling of the wind, and the 
musical example of which is the “ portamento” of the voice or 
that of the violin) is an example of pure melody; pure because it 
is free from rhythmic and harmonic associations. In such a case, 
there is a continuous passage from one arbitrary end-point to 
another ; and both ends may be so shifted as to cover the entire 
pitch-range or any portion thereof. As we reach 435 vibrations, 
for example, we call the tone A, but we shall also call 433, 434, 
436, 437, and more vibrations, A. And so on until a stretch is 
reached over which we are in doubt as to whether the pitch is 
A or Bh. As we approach 460 we are certain of Bh, and A has 
vanished; but the change has been gradual, not abrupt. This 
means that our tone-names are merely convenient points along a 
continuous dimension, any point of which shades imperceptibly 
into its immediate neighbors. Our tones are not separated by 
empty spaces, they are not even clearly defined as points, for 
each has its “ fringe.” 

With this view in mind, we may profitably proceed to an 
enumeration of the most important absolute factors governing 
the melodic relativity of tones. 


4 OTTO ORTMANN 


PART..I 
Static ASPECT OF MELODY 
Absolute Factors 


First and Last Tones—In any succession of sensory stimuli 
the first and the last are accentuated for consciousness because 
the difference between uo sensation and a sensation is greater 
than the difference between any two sensations in the same 
sensory field, so long as intensity remains constant. The first 
and the last tones of a melody, therefore, are projected more 
vividly upon consciousness than any intermediate tone. They 
represent the points where sound breaks silence and silence breaks 
sound. Elaborated into higher mental reactions, this relationship 
Jeads to primacy and recency as efficient factors in recall, the 
former being illustrated by the proverb: “* First impressions are 
lasting.” The first and last tones of a melody mark the end-points 
of the auditory series, and more than other tones, they bound the 
melody. 

Highest and Lowest Tones—In the pitch-series the highest and 
the lowest tones are the psychological equivalents of the first and 
last tones in the temporal series. Above the highest and below 
the lowest tones is a stretch of unstimulated sensory field, while 
on the other side of each there is a sensorial stimulation. Again 
we have the condition of no stimulation against various degrees 
of stimulation. This results primarily from the one-dimension- 
ality of the pitch-series, which forces one of any three tones to 
lie between the other two. ‘Two pitches, therefore, will form 
the extremes in any group, and these extremes, for reasons simi- 
lar to those given for the first and the last tones, are accentuated 
for consciousness. 

This concept is not in conflict with the extensity theory of 
pitch, because the stretch of tectorial or basilar membrane between 
the fenestra ovalis and the point stimulated by the highest tone is 
just as unbroken by other pitch points as that between the lowest 
tone and the apical end of the membrane. On the other hand, in 





ON THE MELODIC RELATIVITY OF TONES 5 


thus combining the highest and lowest tones into a single class 
an impression of psychological equality may be conveyed, which, 
in all probability, is not true to fact. For the volume attribute of 
tones makes any higher tone a “ part of” any lower tone. The 
extensity theory of pitch explains the physiology of this by 
assuming that the same stretch of tectorial or basilar membrane 
innervated for any tone is also innervated for any lower tone, 
plus an additional stretch of membrane. ‘The emphasis for the 
highest and the lowest tones may therefore differ in degree; but 
in any case, both highest and lowest tones are emphasized, in 
comparison with the intermediate tones, as the end-points of the 
pitch-series involved. 

Tone-Repetition—Repetition of any stimulus strengthens its 
impression upon us so long as the repetition is not sufficiently 
pronounced to produce fatigue. Repetition is the sine qua non 
of the Drill Method of instruction, the value of which, whatever 
be its shortcomings, is real. It explains frequency as another 
factor in efficient recall. 1 doubt that true repetition ever 
weakens a preceding impression, even when prolonged sufficiently 
to produce monotony and ennui. What happens in such a case 
is that the new impressions, those of the most recent repetitions, 
are weaker than the earlier ones, and hence may strengthen the 
earlier impression but little, but they do not, therefore, weaken 
the earlier impressions. Constant working with a single pitch, 
carried on long after fatigue or monotony has set in, will none 
the less, often result in the subjects acquiring a memory for this 
pitch. Thus violinists, who otherwise do not possess the so- 
called “absolute pitch,’ not infrequently recognize their one- 
lined A without difficulty, this being the tone from which they 
always tune. For a similar reason orchestral conductors may 
recognize a very slight ‘“sharpness’”’ or “ flatness’”’ of the A of 
the oboe, the instrument from which the orchestra “ tunes,” 
whereas they might not detect these minute differences if the A 
be played on some other instrument. Piano teachers often 
recognize the “ absolute-pitch ” of tones on the piano, but not on 
other instruments. Repetition in these cases has been the chief 


6 OTTO ORTMANN 


factor in retention, and improved retention in simple sensorial 
stimulation means that the original stimuli are summated. 

The degree to which repetition gives emphasis to a tone, de- 
pends, in part at least, upon the number of repetitions. The last 
E’s in the two examples of Fig. 4 make different impressions 
as a result of differences in the number of repetitions. 





A single repetition is not without some effect, ced this 
may then be so slight as to escape notice. 

In any melody the three types of absolute emphasis: the first- 
last tone emphasis (time-extreme emphasis), the highest-lowest 
tone emphasis (pitch-extreme emphasis), and the repetition 
emphasis, are usually all present. The first two must be present 
in every real melody, the third may or may not be present. When 
any two types of emphasis coincide for one or more tones, these 
tones are correspondingly doubly accentuated. If, on the other 
hand, the types of emphasis do not coincide, the psychological 
status of that tone is correspondingly weakened. 





In Fig. 5, the tone E is thrice accented: through pitch-extrem- 
ity, primacy, and repetition. The Ab is accentuated through 
repetition, but weakened, in comparison to the highest and the 
lowest tones, by pitch-intermediacy; while the C is emphasized 
through time-extreme and repetition, and weakened through 


ON THE MELODIC RELATIVITY OF TONES 7 


pitch-intermediacy ; and the Bh is unaccented by any of the three 
factors. Obviously, other combinations are possible, the ultimate 
degree of accentuation depending upon the coincidence or non- 
coincidence of the three types of emphasis. A further modifica- 
tion is caused by a probable inherent difference in the absolute 
degrees of intensity of the various types of emphasis. We cannot 
well compare the degree of emphasis which pitch-position gives 
with that which time or repetition gives, for we are here dealing 
with fundamental sensorial material, which does not lend itself to 
direct comparison. ‘The difficulty is like that experienced when 
we attempt to compare the intensity of a light with that of a tone. 
Accordingly, in addition to the subjective difference in intensity 
between the highest and the lowest tone, resulting from differences 
in tone-form (2), a highest or lowest tone may be per se more or 
less emphatic than a first tone or a repeated tone. 


Relative Factors 


Tones are associated as all sensory data are associated: by 
similarity and by contiguity. In the auditory field, association 
by similarity is the equivalent of contiguity in pitch. Among 
the attributes of tones is tone-form, the resultant of pitch and 
intensity (2). When intensity remains constant, the tone-form 
changes with pitch. Small pitch differences mean great similarity 
of tone-form. On this basis, middle C, for instance, is most 
closely associated with its immediate pitch neighbors, small B 
and one-lined Cz. It is but remotely associated with contra B 
or three-lined Ct. In the first case the pitch difference is but a 
half-tone, in the latter case it is several octaves. The degree of 
association varies inversely as the pitch difference, or pitch 
distance. It is unbroken by any periodicity, such as octave- 
relationship, or fifth-relationship. These are harmonic in nature, 
that is to say, they depend upon reactions to simultaneous tones, 
and are then transferred, in a modified form, into the melodic 
field. The purely melodic association with which we are here 
concerned is made possible by the continuity of the pitch-series, 
and the strength of the association by the complete tonal identity 
which is reached when we make the pitch difference small enough. 


8 OTTO ORTMANN 


Thus, if the pitch difference between two tuning forks be small 
enough, the subject will react to the two tones as “the same 
tone,” which is the equivalent of ideally complete association, 
i.e., identity. A musician will find it difficult to admit the associa- 
tion by pitch-proximity. This is because any tone, to him, at once 
suggests one or more harmonic backgrounds, thereby introducing 
non-melodic elements ; and the same holds for tones of any artistic 
melody. 

- Association by contiguity-in-time causes a tone to be most 
closely associated with its immediate predecessor and successor, 
and remotely associated with tones far removed from it in point 
of time. Here, too, the scale of association is continuous; un- 
broken by metrical symmetry or rhythmic parallelism, both of 
which result from dynamic and temporal variation, factors which 
are excluded from consideration here, because they are the agogic 
equivalents of tonality and scale, and, as such, fall beyond the 
scope of pure melody. 

Interval—W henever two tones are heard in succession at such 
a rate that the impression of the first is still present when the 
second tone sounds, a sensation of interval results. This means 
that a pitch distance between the two tones is consciously recorded. 
In music this relativity is much more important than the recogni- 
tion of any one pitch-point, and may help to account for the 
preponderance of relative-pitch over absolute-pitch. Melodic 
memory (relative-pitch) is a capacity frequently found, whereas 
the existence of tone-memory (absolute-pitch) is relatively rare. 
The normal individual cannot, for example, name the exact tones 
or key of a composition by ear, but has no difficulty in humming 
a short unfamiliar melody immediately after presentation, or a 
familiar melody after delayed recall. 

Accordingly, in an analysis of the reactions to melody, it is 
logical to expect the relative factors to function more often, and 
with greater effect than the absolute factors. Relationships 
among tones will play a more important part than the absolute 
position of any tone, in either the pitch- or the time-series. 

Intervals are associated as separate tones are associated, by 
contiguity in pitch and time. A melodic major third, for example, 


ON THE MELODIC RELATIVITY OF TONES 9 


is most closely associated with a minor third (augmented second), 
and with a perfect fourth; because the pitch distances are most 
nearly equal. It is next associated with a major second and an 
augmented fourth (diminished fifth). Here, again, we must 
guard against letting our harmonic associations (thirds with 
sixths, fourths with fifths, seconds with sevenths) interfere with 
this purely melodic conception. The association of the major 
third with the minor third, for instance, or with the perfect 
fourth, is the closest possible one in our musical system of half- 
tones. But, as in the pure pitch association, many other intervals 
are conceivable between the major third and the minor third or 
the perfect fourth, and these associations would be stronger than 
those given for the half-tone. Again it will be difficult, even 
impossible, for the trained musician to think of melodic intervals 
independently of their harmonic equivalents, but the reality of 
this type of association, too, may be experimentally proved. Thus 
the pitches of two tones may be altered slightly until, let us say, 
a minor third, by imperceptible increments, changes into a major 
third and then into a perfect fourth. We recognize the change 
after a certain stretch of interval has been passed, but cannot des- 
ignate any exact point at which the change takes place ; this part of 
the change being ambiguous, and partaking of the characteristics 
of both intervals. In vision, for example, we may recognize a cer- 
tain distance as a foot. But we also recognize a distance of 12.1 
inches as a foot if measured by the eye alone. After a certain 
ambiguous range we react by describing the distance as more 
than a foot. The case of auditory interval, which is nothing 
else than distance along the pitch-series, is governed by the same 
principle. Since the tones themselves with their physical basis 
are stronger than the interval reaction, we more readily think of 
this type of association in terms of tones (pitch-proximity) than 
in terms of interval. For the same reason we think of three 
unequally spaced points in vision as “two closer together than 
two others.” That is to say, the points rather than the distances 
between them, occupy the focus of consciousness. 

Through contiguity in time, an interval is associated most 
closely with the intervals immediately preceding and following it, 


10 OTTO ORTMANN 


just as a separate tone is associated with the tones immediately 
preceding and following it. The contrast, for example, between 
a major second and a fifth is more noticeable, is reacted to as 
greater, if the two intervals are in immediate succession, than 
if they are separated by other intervals of thirds and fourths. 
In the first case association by contiguity in time ts at its strongest 
point. In the latter case it is weakened because the intervals are 
no longer contiguous in time but are separated by intervals the 
absolute values of which fall between the two. In one case the 
transition is abrupt, in the other case it is gradual. This holds 
for sensations in any field. 

Pitch Direction—The most characteristic attribute of auditory 
sensation is pitch (3), and the most characteristic attribute of 
melody (pitch-succession) is motion. This motion is basically 
described as ascent and descent (2). The idea of opposites herein 
conveyed results from the one-dimensionality of the pitch-series. 
The most fundamental thing that we can say about the pitch of 
two tones is that one tone is higher or lower than the other. And, 
for the same reason, the most fundamental pitch-change in a 
group of tones is a change in pitch direction from ascent to 
descent, or the reverse. This fundamentality is at the bottom 
of the test in pitch discrimination usually given as one test for 
musical talent, in which the subject merely reacts to the second 
of a pair of tones as “higher”’ or “lower” than the first; and 
the relative ease with which the test can be given, from the stand- 
point of comprehension and adaptability, as well as the curves 
of distribution and coefficient of variability which have been ob- 
tained, point to this judgment as a fundamental one. In the 
other sensory fields, similar judgments are equally fundamental. 
In vision, we react to the direction relationship between points as 
soon as we react to the points themselves. One point then is to 
the right or to the left, above or below the other. In kinesthesis, 
one weight is either lighter or heavier than the other. And the 
tendency to project the direction relationship between two points 
in an unbroken or straight line is the explanation of several of 
the well-known optical illusions (3). This tendency, in turn, 
results from the fact that the simplest linear relationship is that 


ON THE MELODIC RELATIVITY OF TONES 1} 


of the straight line. The straight line, in consequence, represents 
the greatest degree of unity. It is not subdivided at all (4). 
Now, if we accept pitch as transtensity, that is to say, as that 
attribute of sensation determined by the amount of stimulated 
membrane affected, the reactions to the auditory direction rela- 
tionships will be similar to those in the visual field (5, 6, 7, 8). 
Accordingly, unchanged pitch direction, the straight pitch-line, 
is the primary basis of melodic tone-association. A succession of 
tones each of which is higher than the preceding tone has an 
attribute common to the entire melody, viz., pitch-ascent. If 
each tone is lower than the preceding tone, pitch-descent is the 
common attribute. As we shall see later, the contour of a melody 
is fundamentally determined by pitch-ascent and pitch-descent. 

In the study of voice-leading, which is the melodic aspect of 
music, one learns three types of tonal motion: parallel, contrary, 
and oblique. In terms of pitch direction these are, respectively: 
ascent or descent; ascent and descent; and ascent or descent with 
horizontality. This classification occurs at the very beginning 
of instruction probably because it represents the most basic 
melodic relationship. And many of the later rules of chord or 
tone succession, which point out the undesirable progressions, 
may be explained on the basis of a change in pitch direction. 
Among them are such well known instances as parallel octaves 
and fifths; the approach to the seventh of the chord; the descent 
of the leading-tone; wide skips; the violation of the natural ten- 
dencies of certain scale degrees; and two fourths or fifths in 
succession in the bass (9, 10, 11, 12). In each case a change 
in pitch direction will eliminate the undesirable feature of the 
progression. 

If straightness of pitch direction is the basis for unifying 
tones, then a change in pitch direction becomes the prime deter- 
minant of melodic tone-groups. A melody ascends, descends, 
or progresses horizontally. In thus moving, it creates melodic 
outline. This auditory outline is most uniform when there is no 
change in pitch direction; it loses uniformity with each change 
of pitch direction. The amount of disjunctiveness thus produced 
depends upon the amount and frequency of change in pitch 


12 OTTO ORTMANN 


direction; small changes in pitch direction will obviously produce 
less disjunctiveness than large changes. However, each change 
of direction, be it great or small, acts as a point of disjunction, 
separating the melodic groups. This grouping, in its simplest 
form is illustrated in Fig. 6. 





Any unnaturalness in this grouping, which the trained musician 
will experience, results from the admixture of group-determinants 
other than that of pitch direction. These will be discussed later. 

Pitch-Proximity—Tones are further associated on the basis of 
pitch-proximity. The nearer the pitches of two tones, the stronger 
is the association. Two tones of widely different pitches are 
dissociated. (Not dis-associated, which is an active taking-apart, 
whereas dissociation means merely the absence of association. ) 
Here again, a comparison with the visual field will make this 
clear. In vision, three points unequally spaced are grouped 2-1 
or 1-2, on the basis of space-proximity, and not 1-1-1. Four 
points, unequally spaced, are primarily grouped 3-1, 1-3, 2-2, 
but not 4-0 or 0-4. The auditory equivalent of such grouping 
is proximity in pitch, and, accordingly, tones are associated on 
the basis of nearness of pitch. Such grouping is shown in Fig. 7. 





The clearness of definition of these groups depends upon the 
ratios existing between the pitch differences of any group and 
the pitch difference between the groups. If, for example, the 
tones within a group are all separated by half-tone steps, and 
the next group begins at the interval of a tenth, the demarkation 
would be very pronounced. If, on the other hand, the tones of 


ON THE MELODIC RELATIVITY OF TONES 13 


a melody are all at the same distance from one another, then no 
grouping on the basis of pitch-proximity can take place. Such a 
condition is shown in Fig. 8. (The grouping, if any, in this 





melody, is the result of a change in pitch direction and of repeti- 
tion.) Between the extremes shown in Figs. 7 and 8, all degrees 
of intermediacy may occur, resulting in a scale of group-definition 
that shades from very marked into zero definition. It is obvious 
that this is but the reverse side of the interval-association, because 
pitch-proximity necessarily means smallness of interval. In view 
of this fact it might be advisable to discard the interval view- 
point entirely. However, certain reactions to tones, which we 
shall have to consider, can be analyzed somewhat more clearly 
by using the interval as a basis. In thus keeping the two distinct, 
I do so solely for practical, not theoretical reasons. Theoretically, 
only one principle is expressed. 

When tones are arranged in the order shown in Fig. 9, a group- 
ing on the basis of pitch-proximuity cannot occur, in spite of the 
fact that the pitch distances vary. Here the change in interval is 
a constant, forming an arithmetic series with either a+ or a —d, 
and thus preventing the formation of any second group. 





If grouping takes place at all, under such conditions, each 
added tone is linked to the preceding, forming either one whole 
group, or shorter groups of two tones each; C with D, D with Ff, 
Ft with C, C with Gg. But such an association is not one of 
pitch-proximity, but of temporal proximity, a form which we 
shall consider later. 

Emphasis in Tone-Groups—Pitch-extreme, time-extreme, and 
repetition emphasis affect the individual tone-group in a manner 


14 OTTO ORTMANN 


similar to that for the entire melody. Thus the first and last 
tones of a group, the highest and lowest tones, and any repeated 
tone within the group become accentuated in relation to the 
remaining tones of the group. These relationships are shown 
in Fig. 10, where the accented tones are marked. 





Interval Relattonship—The existence of interval sensation, as 
distinct from the sensations of the two pitches involved, is as 
real as the existence of distance in the visual field. But if we 
grant the existence of interval sensation, then we must likewise 
grant the existence of interval retention and of interval com- 
parison. Intervals are not only perceived, they are also remem- 
bered and compared. ‘The emphasis, therefore, which the first 
and last tones, highest and lowest tones and repeated tones, 
receive, may be similarly applied to the first and last, highest and 
lowest, and to repeated intervals of the melody. As a result, 
the tones adjoining the first and the last tones, and those adjoin- 
ing the highest and the lowest tones of the melody are accentuated, 
since an interval cannot be separated from its pitch boundaries, 
and as soon as an interval is given, two pitches must be given. 
This connection between pitch and interval makes a separate 
detailed treatment of both superfluous, because a change in pitch 
direction always involves a change in interval excepting the 
change from ascent to descent and the reverse. The conclusions 
drawn under pitch direction may therefore be explained as change 
in intervals also. However, interval introduces emphases other 
than those of pitch direction, and these need to be mentioned. 

On the basis of pitch direction, the emphasis of a tone remains 
unchanged so long as its immediate environment remains un- 
changed. It does not change with the absolute position of the 
tone in the melody. Interval, on the other hand, does introduce 
such a change. In Fig. 11, the D and the Bh are not accentuated 


ON THE MELODIC RELATIVITY OF TONES 15 


Fig.// 


_—— 


either through pitch-extreme or through time-extreme, since in 
both respects they are intermediate tones. But the D and the Bh 
are accentuated because they belong to the first and the last 
intervals of the melody. For a similar reason the tones preceding 
and following the highest and the lowest tones are accentuated ; 
in Fig. 11 these would be Ft and Bh. One must remember, how- 
ever, that the strength of these stresses depends upon the entire 
tonal environment, in which other types of accentuation may be 
so pronounced as to annihilate the interval emphasis. Generally 
speaking, the tones adjoining an emphasized tone, are in turn 
emphasized through interval relationship, though, normally, the 
melodic type of this emphasis is weak. 

Degrees of Emphasts—The theoretical strength of tonal em- 
phasis may be determined either on a basis of change of pitch 
direction or on a basis of interval relationship. In the former 
case, the degree of emphasis depends upon the acuteness of the 
pitch angle. This is best shown by projecting pitch upon the 
vertical dimension of the visual field, and tonal sequence upon 
the horizontal dimension. 





16 OTTO ORTMANN 


The emphasis of any point in the outline of Fig. 12 depends 
upon the acuteness of the angle at that point. A straight angle 
represents zero emphasis, and a zero angle represents maximum 
emphasis. In practice we do not find the latter, because an 
integral part of melody is succession, and succession makes a 
zero angle impossible. The projection shown in Fig. 12, so far as 
linear relationships are concerned, is an adequate counterpart of 
the auditory experience, and the points that “stand out” for 
the eye are exactly those that stand out for the ear. A direct 
comparison between the two is made possible by the extensity 
theory of pitch. However, the absolute dimensions of Fig. 12 
are, of course, arbitrary. 

If we use interval relationship as the basis for determining the 
degree of tonal accentuation, this may be measured by the ratio 
between two intervals. Assuming the pitch range of a melody 
to be approximately that of the average human voice, namely, 
a twelfth, the smallest ratio of interval change, in a system the 
smallest interval of which is the half-tone, would be as 1:19. 
This ratio is shown in Fig. 13. As we decrease the original 
interval, the half-tone increment assumes increasing importance, 
as in Fig. 14. 





This results from the operation of Weber’s Law (13), which 
makes the perceptibility of an addition to, or subtraction from, 


ON THE MELODIC RELATIVITY OF TONES 17 


a stimulus depend upon the ratio of the increment to the base. 
In spite of the modifications which this law undergoes as we 
approach the limits of any sensory series, it is fundamentally 
valid. 

Accordingly, a half-step seems much smaller, for example, 
when preceded by the interval of a twelfth, than when preceded 
by the interval of a whole-step. Its perceptibility depends upon 
the ratios shown in Fig. 14. In this “ seeming-smaller ” is the 
explanation of the association through pitch-proximity. The 
more proximate three tones are within themselves, and the less 
proximate they are to the other tones of the melody, the stronger 
will they stand out as a tonal group. If we turn once more to 
the optical illusions, we shall find the same principle operating 
in many of them (3). In kinesthesis, too, the reaction to any 
absolute weight is influenced by a preceding or succeeding weight 
(24). 

Before proceeding to other types of tonal emphasis, it will be 
advisable at this point to analyze a short melody, combining all 
the types of emphasis thus far considered. The complexity of 
our purely melodic reaction to melody may then be seen. 


OTTO ORTMANN 


18 


‘[BAJOUL JSAMOT pue saysty ‘Js e] pue JsIy == N + 9u0} Jsb] pue ysIG == JN ‘ asueyo 
youd sour Jo ‘drysuonejet [eAtoUl == T ‘uOl}}0d9I-9U0} == Ys "AA JO JUVOSaP OF JUIISE FO 
aZuryo == f {9u0} ysamoy JO ysoysty =] ‘MoT AIBA JO ysiy Arava ‘youd aynjosqe == FY ‘ST “By UT 


N (N) N N(N) NN (N) N (N) 
ct ee aes 8 Yas be Ca) peel 1 bgp hee eg oe ga. Sie tl) ol at ate) = =) 
>, oy 2 sy +» e™ zy i» 2 | iy > » 2 » >» 
; RED ; r f r r 
" i 


| 





ON THE MELODIC RELATIVITY OF TONES 19 


PART II 
TEMPORAL ASPECT OF MELODY 


Thus far melody has been analyzed from what may be called 
its static side. The pitch series alone has furnished the varia- 
tions, while the temporal series has been considered as a constant. 
In actual music, of course, such is not the case. Accordingly, 
we have now to consider the effect of this temporal extent upon 
the types of emphases thus far considered; for our definition 
of melody included not only pitch variation, but also, and that 
necessarily, pitch succession. 

A melody must have duration or extent in time. Inseparably 
interwoven with this extent are the psychological problems of 
immediate memory, recall, and anticipation, because conscious- 
ness never contains only the material in the focus, but also that 
in the fringe of the mental field (14). 

Generally speaking, our reaction to a stimulus diminishes in 
intensity as the time between the presentation of the stimulus 
and any present moment increases. It is this phase of reaction, 
as we have seen, that denominates recency as a factor in vivid- 
ness of impression. The basis of this vividness is found in the 
association by contiguity, which operates in the agogic series as 
well as in the pitch-series, and causes any tone to be most closely 
associated with its immediate successor, and less closely associated 
with tones further removed in temporal extent. 

First and Last Tones-—The emphasis which, by virtue of these 
positions in the melody, the first and the last tones of a melody 
receive is counteracted, so far as the first tone is concerned, by 
the temporal factor. Since the first tone is always the tone 
farthest removed from the objective presentation of other tones, 
its projection in consciousness is weakest. If the melody be 
sufficiently long, that is to say, if it quite exceeds the memory- 
span, the first tone, in spite of its original emphasis, will be lost 
for consciousness. The last tone, on the other hand, receives a 
double emphasis, because to the accentuation of boundary tone 


20 OTTO ORTMANN 


of the sensory series, is added the accentuation of greatest re- 
cency. Immediately after the completion of a melody, therefore, 
the first tone has passed through all stages of agogic emphasis, 
from maximum to minimum, while the last tone stands at maxi- 
mal emphasis. 

Highest and Lowest Tones—In a similar manner the original 
stress which the highest and the lowest tones of a melody receive 
through their positions in the pitch-series, gradually diminishes 
as the melody proceeds, unless it is reénforced through repetition, 
until it, too, may be replaced by the recency-emphasis of some 
pitch-intermediary. And of the two tones, that one having the 
greater recency, other things equal, will be the more emphatic. 
But this effect works also the other way. If the difference in 
pitch-extreme accentuation originally be great enough, it will 
overcome the recency-emphasis just referred to. In Fig. 16, in 
spite of the greater recency of the low Bb, the very marked pitch 
accentuation of the high Git may suffice to retain the latter tone 
in consciousness more vividly than the former tone. 





Here again the greatest emphasis results when the two types 
of stress coincide; when, for example, the last tone is also the 
highest tone. This is very characteristic of many dramatic 
melodies. One need only recall, for example, the “ Eri Tu” 
from Verdi’s Masked Ball, the ‘‘ Celeste Aida,’ the close of 
Tristan and Isolde, the optional end of the Lucia Sextette, or 
Mendelssohn’s “ Hear Ye Israel.”” When this coincidence is 
absent, the accentuation for any one tone is correspondingly less. 

Tone-Repetition—Considered in its temporal aspect, the em- 
phasis which a tone receives through repetition depends, first, 
upon the time-interval between the repetition and the original 
presentation of the tone; secondly, upon the number of repeti- 
tions; and, thirdly, upon the pitch-emphasis status of the tone 


ON THE MELODIC RELATIVITY OF TONES 21 


at any presentation. That it depends upon the length of time 
interval may readily be shown by giving a subject a series of 
discreet tones, in which the repetition of a given tone is to be 
noted. As the time-interval increases the recognition will become 
more and more uncertain until, finally, it is entirely lost. A pre- 
liminary survey of this difficulty has shown that, with a single 
presentation of the tone, and the series played at the rate of one 
tone per second, very inferior pupils “ forget’”’ the tone after 
seven seconds, while superior pupils (not possessing the so-called 
absolute-pitch) recognize it after 21 seconds or more. It is this 
variation that forms the basis of many memory tests (15, 16). 
Variations in the types of emphases discussed in Part I, of course, 
can modify this relationship between difficulty and length of time- 
interval. 

The dependence of stress upon the number of repetitions, pro- 
vided that they be recorded as repetitions by the subject, has 
already been mentioned. It is the “ frequency’”’ emphasis, and 
is not complicated by temporal considerations other than that each 
repetition is subject to the variations mentioned in the preceding 
paragraph. 

The third element in repetition-emphasis, however, is less 
simple. It depends upon the pitch status of each presentation. 
In Fig. 17, the repetition of C will be recognized more readily 
in 1 than in h, because in 1 the second C is already emphasized 
through change of pitch direction and size of interval, whereas 
in h the second C is unemphasized in these respects. 





For a similar reason, the tone C, Fig. 18, 7, is more accentuated 
finally than the C in h, because in the former each presentation 
of the tone is stressed through lack of pitch proximity with ad- 


22 OTTO ORTMANN 


joining tones, whereas in / pitch-proximity tends to obscure the 
individuality of the tone at each presentation, linking it more 
closely to the other tones of the melody. 





Thus the mere fact that a tone has been repeated four times 
does not mean that the fifth repetition will be equally strong in 
all cases. The degree of final emphasis can be determined only 
if we know the tonal environment of each presentation of the 
tone. 


The influence of the temporal series upon the psychological 
status of any tone of a melody is seen clearly when we construct 
a melody tone by tone, which is the way in which it is heard, if 
we exclude the play of anticipatory imagery. In Fig. 191, the 
F¢ is accentuated as last and again as highest tone. In 7, it re- 
mains emphasized through change of pitch direction. In k, this 
is lost, because of the higher Gf. In Jl, the F# is reénforced 
through repetition; in m, it is weakened as intermediate tone ; and 
in , it is again reenforced by repetition. 











y- 
Ad9 4 Heer oR ve —-—_ F#o—H, a; ae 
a> 





Each tone in any melody undergoes similar changes. It fol- 
lows that the status of a tone in consciousness is not controlled 


ON THE MELODIC RELATIVITY OF TONES 23 


by any one series, either pitch or time, but is modified by the 
tonal environment. The same conclusion applies to any group 
of tones, and, incidentally, to any chord or group of chords (17). 

Thus the psychological status of any tone of a melody is not a 
constant. 

Higher-Units—Association of tones into higher-units is 
strongest when both types of association—contiguity-in-pitch and 
contiguity-in-time—are present for the same group of tones. In 
Fig. 20 h, the tones can be associated only through contiguity in 
time, since the interval relationship (wide pitch difference) makes 
pitch association (not harmonic association) impossible, or at 
best weak. In Fig. 201, the reverse is true. In this example, 
according to association in time, the tones would be linked: 
1—2—3-4—-5-6-7-8-9. But the- marked pitch-proximity among 
tones 1—3—5—7—9 and among 2-4-6-8 is sufficiently strong to 
replace the time-grouping, so that the ear “hears” the two 
melodies (outlines indicated by the dots). 





In Fig. 20 7, the pitch-proximity between any two tones of the 
melody is sufficiently great to make a division of the tones into 
two melodies impossible. Such an analysis and synthesis are being 
constantly used in listening to music. As a result thereof, objec- 
tive pitch-descent may become psychological pitch-ascent, and vice 
versa. For, in associating tones 1 and 3, we bridge over the 
tone 2, otherwise tones 1 and 3 could not help to form a melody 


24 OTTO ORTMANN 


(interval). The descending major tenth (Fj to D) is replaced 
by an ascending major second (Ff to Git); and the ascending 
eleventh (D to Gi), by a descending second (Dto C). One effect 
of such a substitution is that a violently jerky melody with its 
resulting emotional restlessness, may thus be replaced by two 
smooth melodies with their opposite emotional counterpart. 

The strength of this type of association is determined, on the 
one hand, by the ratio between the pitch association and the time 
association, and, on the other hand, by the relationship between 
the intervals of either melody and the intervals between the two 
melodies. In Fig. 21, poor association in time weakens the upper 


Frg¢. Zh 
—fy——# as -% bel « a 
9) Ak SE ES ER OED EA LARTER BPE DS ORIN = REL TR SRE RE AYR 
St = 
@ —peee? L Pe THe’ © ¢ cata * — * Co 7? 


melody considerably but not the lower. In Fig. 22, poor associa- 


fig 2b 
e 2 . 2 
4 : : 7 ie £2 
((, een. SCARE ESMMNRMONINSENET a NaC td e POF CRPATE Bs pee TDS Ai BO 
g s- & | sale 


tion in pitch makes a division into two melodies impossible, since 
the intervals of either intended melody would be as great as the 
intervals between the two melodies (major third or augmented 
fourth and diminished fifth). The tones Ah can equally well 
belong to the higher or lower melody. The smaller this ratio 
between the intervals of a melody and the intervals between two 
implied melodies, the more pronounced are the individualities of 
the melodies. In Fig. 22, the ratio is 1:1, at which point no 
grouping whatever takes place. (Contrast this with the clear 
definition in Fig. 20 1.) 

Tempo—Intimately connected, of course, with all considera- 
tions of temporal relation is the question of tempo. This does 
not introduce any new elements, but does modify the relation- 


ON THE MELODIC RELATIVITY OF TONES 25 


ships which we have discussed, by increasing or decreasing the 
absolute time value of any tone or any group of tones. By so 
doing, it may increase or decrease the memory-span, and thus 
alter the size and quality of the tonal higher-units. We are here 
very close to rhythm, for the law upon which such grouping is 
based is at the same time the fundamental law of rhythm: that 
short tones are more strongly grouped than long tones. In this 
connection the deductions and examples of Parts I and II pre- 
suppose a constant tempo. 

Intensity—A further important element is intensity. ‘Treat- 
ment of its effects, like that of tempo, is here omitted because 
the aim of the discussion is an analysis of the purely melodic 
(pitch) aspect of melody. Both tempo and intensity are men- 
tioned, however, because either may readily overthrow the re- 
actions which have been enumerated. A tone unemphasized as 
pitch intermediary, for example, may overbalance the highest 
or the first tone, if it be given intensity accentuation. In fact, 
since intensity is the sensorial reaction to the dimension directed 
toward the end-organ (the degree to which an end-organ or 
group of end-organs is affected), and has, therefore, greater 
biological significance, it is probable that a slight increase in 
intensity will outweigh a considerable increase in pitch or time 
emphasis. ‘The deductions of Part IV also presuppose the same 
careful separation of the rhythmic from the purely melodic 
elements of melody. 


26 OTTO ORTMANN 


PART III 
INFLUENCE OF EXTRANEOUS FACTORS 


The various types of tonal relationship which we have con- 
sidered color or modify our reactions to any series of single 
tones. In music these reactions are further modified by har- 
monic and rhythmic influences, both of which are to be excluded 
here as far as possible. Complete elimination, however, is im- 
possible, for reaction to melody is a psychological phenomenon, 
and the integrative action of the organism excludes complete iso- 
lation of any one factor. In the first place, it is impossible to 
select an example of pure melody on any musical instrument 
tuned to the tempered scale. Pure melody must be an atonality 
melody, which means one outside the tonality scheme (not anti- 
tonality). As soon as we introduce the smallest fragment of key 
or tonality, we introduce harmonic imagery. Now, there is no 
interval in the tempered chromatic scale which is not found, or 
the enharmonic equivalent of which is not found, as an interval 
of our diatonic scale. The augmented-fourth, even, occurs be- 
tween the fourth and the seventh degrees of the scale and im- 
mediately suggests, among other things, resolution into the third 
and eighth degrees, as a result of harmonic associations (domi- 
nant-seventh-chord into tonic-triad). The first two tones of 
any melody, regardless of the interval between them, may, and 
most often do, suggest tonality relationships, by being heard as 
certain tones of an imaged scale, which may or may not counter- 
act the purely melodic relationship between the two tones. The 
same thing holds for any other tone or group of tones in a melody. 

Now these intervals, through differences in the frequency of 
their use and their harmonic importance, vary in their associative 
importance. A second or a third is much more vivid as an 
associative link, that is to say, either interval awakens tonality 
associations more readily than the augmented-fourth, because, 
among other things, the former are more frequently used in 
melody than the latter. If we select any good melody, we shall 


ON THE MELODIC RELATIVITY OF TONES 27 


find the melodic steps of an augmented fourth or a diminished 
fifth less used than those of seconds and thirds. In each of the fol- 
lowing examples, which include all possible cases for a whole 
tone octave, tonality associations function: C-D; C—E; C-F¥; 
C-—Git; C-Ag. On a purely melodic basis, the strongest associa- 
tion is the diatonic one, since this also represents the greatest 
tone-similarity through pitch-proximity, excluding .only tone- 
repetition. It is interesting to note in this connection, that dia- 
tonic progression excuses many otherwise forbidden harmonic 
progressions, such as the violation of the natural tendencies of 
active tone-steps (18), successive doubling of the bass in sixth 
chords (19, 20), and the II-V-I progression with the 6-7-8 
soprano (21), rules with which students of harmony are familiar. 
Next in importance is the interval of the third, since it represents 
the second of the two fundamental harmonic relationships (25). 
For present purposes we may limit it to the major third, because 
the minor third is excluded from the whole-tone scale, which has 
been adopted here in order to eliminate tonality associations as 
far as possible. And, since chords are built in an ascending 
direction, we may expect to find the strength of association for 
the third differing with the direction of the interval. An ascend- 
ing major third will awaken stronger tonality associations than 
a descending third, other things equal (22). The order of the 
other intervals in regard to their associative importance is not 
so clearly determined, since the frequency of their use varies 
considerably. 

In order to secure some concrete data on the actual frequency 
with which the various intervals are used in artistic melody, a 
count of one hundred and sixty songs was made, on the assump- 
tion that composers in selecting tonal sequences for melodies must 
follow melodic relationships, and also that frequent presence of 
certain intervals, in turn, strengthens the association between 
the tones of these intervals for the hearer. The composers 
selected were four of the best song composers, Schubert, Schu- 
mann, Brahms, and Richard Strauss, and the works chosen con- 
tained most of their best songs and some less well known. Ap- 
proximately twenty-three thousand intervals were counted. This, 


28 OTTO ORTMANN 


of course, would be too small a number to warrant generalizations 
if the various coefficients of deviation and dispersion showed wide 
variation. But the opposite is the case. Thus unisons or seconds 
were first in frequency in 9714 per cent. of the songs, thirds 
leading in the remaining 2% per cent. The frequency orders: 
unison, second, third, fourth; or second, unison, third, fourth, 
were found in approximately 60 per cent. of cases. Such uni- 
formity gives considerable general value to the count as made, 
and reveals clearly the presence of a melodic relationship of fixed 
order. 

The first of the accompanying two figures (Figs. 23 and 24), 
shows the percentile distribution of the presence of each interval 
in the songs taken as a whole; that is to say, the number of songs 
out of the total in which the interval was present once or more 
often. The second figure shows the percentile distribution of the 
intervals themselves, thus taking into account the actual number - 
of times in each song that the interval was present. 

The figures are to be interpreted as follows: If the melodic 
relationships outlined in the preceding pages actually exist and 
operate in artistic music, then, in melodies (songs) small inter- 
vals should predominate markedly over wide intervals. This 
they should do regardless of their harmonic (fusional) value. 
The interval of a second, although a dissonance, should occur 
more frequently in a melody than a third, although the latter 
is a much better harmonic interval. A seventh, the harmonic 
equivalent of a second, should occur rarely in a melody on 
account of the remote pitch-proximity between its tones. Thirds 
and sixths, considered generally interchangeable from a harmonic 
standpoint, should show a marked preference in frequency for 
thirds, whose melodic relationship is more than twice that of 
sixths. A more pronounced form of this difference should be 
found for unisons and their harmonic equivalents, the octaves. 
Between fourths and fifths the difference will be less, because 
the melodic relationships of the two intervals are more nearly 
equal, but should still be in favor of fourths. 

In Fig. 23 the essential features of melodic relationship, as 
found in the melodies selected, are shown. Distances along the 


ON THE MELODIC RELATIVITY OF TONES 29 


axis of ordinates represent the number of songs in per cent. ; 
distances along the axis of abscissas represent the intervals. In 
this figure only the most frequently used intervals of the diatonic 
scales are included because the curve is intended to show only 
the central tendency of melodic relationship. 


: Fro. 23 
loo 
40 
80 
70 
60 


50 


Two deviations from the expected distribution, one slight, the 
other more marked, need explanation. The increase of fourths 
over major thirds is the result of the operation of harmonic 
relationships, the fourth (as inverted fifth) playing an important 
role in our harmonic system. The increase of the octave over 
sevenths shows this more clearly. Harmonically the octave is 
the most fundamental interval (except the unison) whereas the 
seventh is a dissonance. Melodically the two intervals have 
almost the same relationship, since the pitch-proximity of their 
tones is almost the same. The harmonic value, therefore, out- 
weighs the melodic. 

In Fig. 24 the dotted line reveals what may be called the ideal 
curve of distribution for melodic relationship. This is based 


30 OTTO ORTMANN 


entirely upon pitch-proximity, the major second having one-half 
that of the minor second; the minor third one-third that of 
the minor second; the major third, one-fourth; and so on. The 
starting point for the upper end of the curve was chosen midway 
between the frequency of the unison and minor second found 
in the test as made. (Since the form of the curve remains the 
same, its absolute position in the figure is immaterial.) The 
solid line shows the distribution as actually found. Vertical 
distances are per cents of frequency; unisons, for example, con- 
stituting 26 per cent of all the intervals counted, minor seconds 
17% per cent., and so on. 


%o 
26 
24 


22 





ae 


A 
ae), MRA 2) Neeoitaae bo ca 


The agreement of this curve with the theoretical curve is well 
marked, and if the real curve were presented in the form of a 
smoothed histogram, the agreement would be striking. The dif- 
ferences between the two curves may be explained as follows: 


ON THE MELODIC RELATIVITY OF TONES 31 


Since all songs selected were written in the tonality system of 
the music of Western Civilization, harmonic influences cannot be 
entirely excluded, particularly since the songs were written with 
piano (harmonic) accompaniment. This accounts for the pre- 
dominance of the octave over the sevenths. Major seconds ex- 
ceed minor because in our diatonic scales (the form of scale 
upon which the songs are based), the ratio of major to minor 
seconds is 5:2. Actually, in melody, this ratio is considerably 
reduced through the greater pitch proximity of the minor second. 
In the count as made the ratio is 96:70. Part of the reduction, 
of course, can be accounted for by the presence of the harmonic- 
minor tonality, in which minor and major seconds, from the 
standpoint of scale construction, each occur three times. 

Tonality, or harmonic relationship, also explains the drop in 
the frequency of the augmented fourth. This interval occurs 
but once in our scale, and forms a dissonance which does not fit 
into chord structure readily. Its inversion, the diminished fifth, 
is found somewhat more frequently, and accounts largely for 
the percentage at this point of the curve. Without the diminished 
fifth, the augmented fourth would fall to one-fifth of one per 
cent. With the diminished fifth included it is still below the 
frequency of sixths. 

The presence of melodic relationship is best seen in comparing 
harmonically equivalent intervals: unisons with octaves, seconds 
with sevenths, thirds with sixths, fourths with fifths. Intervals 
that have the greatest harmonic relationship then have the least 
melodic relationship, a relationship shown by the difference in 
height between the two, Fig. 24. As the melodic relationship 
increases, that is, as the pitch difference decreases, the frequencies 
of the intervals in question approach a common level. (Between 
unisons and octaves the difference equals 25 per cent.; between 
seconds and sevenths 20.7 per cent.; between thirds and sixths, 
7 per cent.; between fourths and fifths, 6 per cent.) The actual 
percentages are variable, but the general decrease results from 
the operation of the basic principle of melodic relationship. 

Finally, by combining the curve for melodic relationship with 
one for harmonic relationship, as in Fig. 25, any doubt as to the 


32. OTTO ORTMANN 


opposition of the two forms of tonal association, the vertical 
and the horizontal, will be dispelled. In this figure the solid line. 
represents the consonant (harmonic) value (equal temperament), 
of each interval (taken from Helmholtz, 26), and the dotted line 
the melodic value as found in this test. Where the dotted line 
is above the solid line melodic associations predominate, where 
the dotted line is below, harmonic values predominate. The shaded 
portion marks the degree to which the two forms of association 
do not coincide. This, it will be seen, covers a great part of the. 
entire area of the figure, thus proving the low correlation between 
the harmonic and the melodic principles of tonal association. 





In like manner, rhythmic influences alter the melodic relation- 
ships which we have enumerated. The exclusion of objective 
intensity differences does not necessarily. destroy subjective 


ON THE MELODIC RELATIVITY OF TONES 33 


rhythm. A series of absolutely even clicks, both as to intensity 
and to duration, may very easily be heard as groups of two, 
three, four, or more. Accordingly, any tones that happen to 
fall on a subjectively accented beat, will receive a rhythmic or 
metrical accent that may counteract a lack of purely melodic 
emphasis; or, on the other hand, they may strengthen a melodic 
accent already present. Since the subjective rhythm supplied by 
the listener may readily differ between two subjects, these 
rhythmic accents may fall on various tones for the same melody. 
These rhythmic influences cannot be entirely eliminated, but by 
keeping the intensity constant, and the rate of tonal succession 
sufficiently slow (one tone or less per second), they can be mate- 
rially reduced in importance. 

_Tonality and intensity influences are harmonic and rhythmic 
problems respectively, and, as such, they fall beyond the scope 
of the present article. They have been briefly mentioned, how- 
ever, because without them, certain discrepancies which appear 
in all tests for melodic reaction when compared with the un- 
analyzed reaction of the mature musician cannot be adequately 
explained. It is also advisable to mention, here, the influence of 
the anticipatory judgment (14). The preparation of new stimuli 
through anticipation may result in an added emphasis, if the 
stimulus coincides with the anticipation, and may require a re- 
adjustment, with its resulting confusion, if it does not. Through 
anticipation, a tone otherwise not emphasized, may be stressed 
for consciousness; and an emphasized tone may be weakened. 


34 OTTO ORTMANN 


PART IV 
Errects oF MELopic RELATIONSHIP 


As a result of the predominance of relative over absolute pitch, 
and of the constantly operative psychology of the higher-unit, 
it is impossible, objectively, to change a single tone of a melody 
without changing the psychological status of its adjoining tones, 
and, to a less degree, that of all other tones of the melody falling 
within the memory-span. 

Thus, if the following two motives be given for comparison, 
and the subject be required to judge whether the first, second, 
or third tone has been changed, the last tone of the second motive 


is sometimes judged as different on account of the change of 
interval between F{—-E and Gi#-E. 


Fig.26 
Se 
Conversely, in the following examples, the last tones are some- 


times heard as the same tone since the last interval (descending 
third) is the same in both examples, Fig. 27. 


Fig.27 


= 


Such an error may function regardless of the actual pitches 
used. In groups of only three tones, which are well within the 
memory span of the normal pupil, this type of error is seldom 
found. In longer groups, however, it is more frequently 
observed. 

The effect of changes in pitch direction, from ascent to descent 
or vice versa, also introduces an error of judgment in tonal com- 


ON THE MELODIC RELATIVITY OF TONES 35 


parison. If we compare the two motives of Fig. 28, the third 
tone (Ah) is frequently heard, in addition to the second tone, 
as altered tone. This judgment seems to be based upon a change 
in pitch direction. A descending interval with its judgment of 
“lower than” is replaced by an ascending interval with its judg- 
ment of “ higher than.” 


Fig. 28 





The melodic line here has been changed from ascent-descent- 
ascent, to ascent-ascent-ascent, a change sufficiently basic to 
destroy, or, at least perceptibly weaken the individual status of 
each tone. If such a change involves, at the same time, a change 
from and into the highest or the lowest tone, the error is further 
intensified. 

In comparing the two examples of Fig. 29, a more involved 
operation of this type of error may be seen. Here the fourth 
tone, in addition to or in place of the correct second tone, may be 
reacted to as the changed tone. This results from the association 
of tones two and four through pitch-proximity, which the inter- 
vening low third tone does not entirely destroy (see Fig. 20). 
In the first of the examples, the second tone is higher than the 
fourth tone, in the second example, the fourth tone is higher than 
the second, which results in an interchange of pitch direction 
similar to that outlined for Fig. 28. This error is more marked 
if we omit the upper E, and thus make the change of tones also 
a change of highest pitch, C to Bb. 





The effect of interval relationship is seen when we alter two 
tones equally in different environments. For example, in an 


36 bis OTTO ORTMANN 


environment consisting only of thirds, a change of a whole-tone 
seems larger than the same change in an environment of larger 
intervals. This difference is shown in Figs. 30 and 31. In 
the first example, the change from C to D is one-half of the 
interval E—C, and one-half of the interval D-Bh. In the second 
example, the change from Bb to C is also a whole-tone, but this 
distance forms only one-fourth of the interval D-Bh, and one- 
fifth of the interval Bh—-Gf. 





This ratio of increment-of-change to base, seems to be one of 
the determinants of the judgment. For a like reason, a single 
large interval in a series of small intervals, is more noticeable 
(emphasized) than the same large interval in a series of larger 
intervals. The high Gt in Fig. 21, is an illustration of this 
emphasis. 

The element of repetition affects tonal judgments by producing 
a judgment of sameness versus difference of pitch, which is more 
fundamental than a judgment of difference in a series of differ- 





ences. Thus in Fig. 32 the change D to C results in a repetition 
of the tone C, as a result of which the second C receives repe- 
tition emphasis, which automatically emphasizes the alteration 
itself. In Fig. 33, no repetition through the alteration occurs, 


ON THE MELODIC RELATIVITY OF TONES 37 


although the quantity of change remains the same. The altera- 
tion in the second part, other things equal, is therefore somewhat 
more difficult for the listener to detect. 





Finally, other things equal, the vividness of any change in a 
melody depends upon the length of the melody. Of two equal 
changes, made in a short and in a long melody, the former will 
always be more readily perceived than the latter (23). 


38 OTTO ORTMANN 


PAR DW 
A Test In MEtopic MEMORY 


The usual test in music for melodic memory, combines the 
memory-span method and the method of recognition. The span 
method demands that we begin with a series of auditory stimuli 
quite within the limit of the subject and that we increase the 
series until errors begin to occur, meanwhile keeping other factors 
constant. The method of recognition demands the representa- 
tion of an altered series in which the subject points out either the 
altered or the non-altered stimull. 

The modest demands which such a test seems to make, at first 
acquaintance, are replaced by insurmountable difficulties upon 
closer analysis. For, if the theoretical analysis of the preceding 
pages be correct, then it is impossible to “ keep other factors 
constant,’ in any altered series. 

The test here outlined was given to 128 music students, roughly 
divided into two classes. Class A consisted of 89 pupils, mostly 
children from nine to seventeen years, but containing also a few 
adults. Class B was an adult group (over seventeen years) of 
39 pupils. In both classes all degrees of musical talent (measured 
both by teachers’ estimates and, by other more detailed tests) 
were represented, although, since all were students at a music 
conservatory, the entire group probably represents a slightly 
favorable selection. The method used was the memory-span plus 
the recognition methods. Reproduction, in which the subject 
reproduces the stimulus series, was not used for three reasons: 
it makes group presentation impossible; it adds an important 
vocal difficulty ; and it makes actual recording and interpretation 
very difficult, if not impossible. Series of two, three, four, five, 
and six tones were given and five examples in each series were 
used. The subjects were provided with printed forms carrying 
a number for each tone, and they were instructed to draw a short 
dash through any tone that was changed. If, for example, a 


ON THE MELODIC RELATIVITY OF TONES 39 


stimulus of three tones had been given, as model, the subjects 

followed the figures 1, 2, 3; on the form, and crossed out the 

particular tone or tones heard as altered in the comparative 
stimulus when compared to the model. The element of writing 
involved in this procedure introduced no appreciable difficulty, 
and was found more reliable than the method which demands that 
the subject remember the tone until the entire melody has been 
given, and then write the number of the altered tone or tones. 

The examples were given on a small portable organ, at uniform 
intensity, and at the rate of one tone per second with second 
silences between. This assures elimination of legato, or any over- 
lapping of tones, which might readily introduce important difh- 
culties. The five examples of the easiest (two-tone) melodies 
were given before the step to the three-tone melodies was made. 

And this plan was followed throughout the test. Preliminary 

instruction and examples assured proper understanding of the 

test. No rest periods were introduced, since the test consumes but 
little time, thus making allowance for fatigue unnecessary. 
The melodies used were selected on the following basis: 

1. The whole-tone scale was used in order to minimize tonality 
associations. The use of the half-tone as basic interval, that 
is to say the basing of such a-test on the chromatic scale, is 
less satisfactory on account of the important part which the 
half-tone (as leading tone, for example), plays in our music 
system. 

2. All changes involved a constant pitch difference: one whole- 
tone; the smallest amount possible in the system used. 

3. Changes of pitch direction were kept constant. They occurred 
either in none of the examples of any one series or in all of 
the examples (one exception). 

4. The ratio of changed interval to adjoining interval was kept 

constant for any one example. 


Table I shows the percentile distribution of correct answers 
(the per cent. of times that the altered tone and no other was heard 
as altered), for Class A, and Table II, that for Class B. 


40 OTTO ORTMANN 


TABLE I 
2 3 + 5 6 
Example tones tones tones tones tones 
1 84 75 83 58 
2 98 76 57 53 15 
3 96 85 69 47 56 
+ 93 80 65 49 41 
5 97 96 90 49 41 
TABLE II 
2 3 + 5 6 
Example tones tones tones tones tones 
1 100 94 81 79 43 
2 100 87 58 38 28 
3 97 SP ithiis 84 43 48 
+ 100 89 74 38 30 
5 100 93 76 48 41 


The average distribution for the two classes is: 


2 3 4 5 6 
tones tones tones tones tones 
Class A 93 83 70 56 
Class B 99 92 75 49 36 
Ceneechi ny aman 87.5 72.5 52.5 38 


The range for the two classes is: 


Z 3 4 o 6 

tones tones tones tones tones 
Class A 10 20 33 36 43 
Class B 3 10 26 41 30 


This distribution of average correct replies shows the increase 
in difficulty with the increasing importance of relative factors as 
the memory-span increases. As we increase the number of tones 
in a melody we multiply the possibilities of relationships with 
other tones, thereby increasing the chances of tonal emphasis, 
positively or negatively. Since the attempt to keep all factors 
constant, except the changed tones, applied to all examples, the 
ideal distribution would show the same percentage for all 
examples of any one vertical group. We have now to explain the 
deviations from this distribution which the actual results show. 

The deviations in the two-tone columns are too slight to 
demand analysis. The nearness of correct replies to 100 per cent. 
indicates that in the two-tone series we have the practical lower 
limit of tonal memory. 


ON THE MELODIC RELATIVITY OF TONES at 


In order to facilitate the analysis and avoid needless repetition 
J shall select certain examples which deviate sufficiently from the 
average distribution to leave no doubt as to the influence of tonal- 
relationships. Where the deviation is slight, there is danger of 
misinterpreting the distribution, since the number of pupils tested 
is small, and the melodies are too short for the isolation of ary 
one type of tonal emphasis. Consequently such examples are here 
excluded. 

The first question that arises is whether the distributions are 
the result of chance conditions operating for the particular class 
tested, or the result of the nature of the test itself. In order to 
answer this, the test was given a year later to another group of 
adult students, Class C, from sixteen years up. The number of 
times each unchanged tone of each example was heard as changed 
(which represents a loss of imagery for that tone) was calculated, 
as well as the number of times the actually changed tones were 
heard as changed. If there were no influence of extraneous con- 
ditions, the two classes should show very similar distributions as 
far as tonal-relationships are concerned. The average percentile 
distributions, that is to say, the deviation from perfect replies, for 
the four-, five- and six-tone columns, for the two classes were: 


4 5 6 

tones tones tones 
Class B 8 11 Aes | 
Class. G 11 12 12 


These numbers are the percentages of times that tones were 
incorrectly heard as changed tones, that is, the number of times 
that the memory for the tone was lost. The close agreement 
between the two classes, which was, incidentally, accompanied by 
close agreement in the range as well as in deviation for: any 
one tone, shows that the constant factors of the test are being 
revealed, and that any marked deviation, therefore, is indicative 
of differences in the various parts of the test itself. Since about 
10 per cent. represents the average error, a figure considerably 
less than this would mean that that tone must be well emphasized 
for consciousness else it could not be so readily and correctly 
retained. On the other hand, any figure considerably above 10 


42 OTTO ORTMANN 


per cent. indicates that the corresponding tone is unemphasized, 
as a result of which the memory of it is lost. The selection of the 
following examples was made on this basis. Only tones were 
selected upon which the distributions for the three classes of 
pupils agreed sufficiently, and which were sufficiently removed 
from the average to minimize doubt as to the influence of tonal 
relationships. “Tones accompanied by a low percentage of error 
are tones upon which the various types of tonal emphasis that we 
have considered, coincide, or upon which there is a particularly 
strong single emphasis. Tones yielding approximately 10 per cent. 
error, are tones emphasized in certain respects and unemphasized 
in others, constituting what we might call normal emphasis. 
Tones accompanied by high percentage of error are largely unem- 
phasized, at least in comparison with other tones of the same 
melody. The memory error is the per cent. of times that the 
memory for the tone was lost. 





Tone 1. Accentuated as first and lowest tone; unaccentuated as 
a member of the group F-G. (The stronger the 
grouping, the more does the individuality of each tone 
merge into the whole.) Memory for this tone lost in 
13 per cent. of cases. 

Tone 2. Unaccentuated as pitch intermediary. Memory error, 
19 per cent. 

Tone 4. Accentuated as last and highest tone; unaccentuated 
through grouping with the preceding C#. Memory 
error, 7 per cent. 





Tone 1. 


Tone 3. 


Tone 5. 


ON THE MELODIC RELATIVITY OF TONES 43 


Accented as first and highest tone. (Also through size 
of first interval.) Memory error, 3 per cent. 
Accented slightly as first tone group E-D. Unaccented 
as pitch intermediary. Identity confused through result 
of change of interval between it and the real changed 
tone. Memory error, 22 per cent. 

Accented as last tone and through isolation from pre- 
ceding tone on account of wide preceding interval. 
Unaccented as pitch intermediary. Memory error, 13 
per cent. 





Tone 1. 


Tone 2. 


Tone 4. 


Accented as highest tone, as first tone, and through size 
of first interval, which clearly isolates the tone. This 
is a case of coincidence of emphasis on one tone. 
Memory error, 2 per cent. 

Unaccented as pitch intermediary and through prox- 
imity to F. Memory error, 23 per cent. 

Accented as lowest tone, but identity weakened through 
change in preceding interval. Memory error, 9 per cent. 


Tone 5. Accented slightly as first tone of group A—B on account 


of size of interval (augmented fifth, largest interval in 
this melody). Unaccented as pitch and time inter- 
mediary. Memory error, 15 per cent. 





Tone 1. 


Accentuated as first tone; as lowest tone. Accentuated 
as detached tone from the group E-F¥. Triple empha- 
sis. Memory error, 1 per cent. 


44 OTTO ORTMANN 


Tone 3. Unaccentuated as pitch intermediary. Confused with 

the changed tone through the change in the ene 
: interval. Memory error, 16 per cent. 
Tone 4. Accentuated. as highest tone. Accentuated through 
interval relationship with other intervals of the melody 
(diminished fifth compared to major third and second), 
slightly unemphasized as agogic intermediary. Memory 
error, 3 per cent. In this case the emphasis through 
size of interval (pitch isolation) is sufficiently great to 
overcome the non-emphasis through temporal inter- 

mediacy. 

Tone 5. Unaccented as intermediate tone. Slightly accentuated 
through change in pitch direction. Memory error, 12 
per cent. 

Tone 6. Unaccented as pitch intermediary, both for entire 
melody and for second group: C—G# Af. Accentuated 
as last tone. Memory error, 14 per cent. 


If we average the memory errors for tones upon which the 
time- and the pitch-extreme emphases coincide, we get an average 
of 4.5 per cent. for all five- and the six-toned examples. The 
average for tones upon which these two types of emphases do not 
coincide is 13.5 for the same groups. In other words, it was 
noticeably easier to remember a tone, as here presented, when the 
tone was accented both as first and highest or lowest tone, than 
when it was accented as one but not at the same time as the other. 

The effect of interval memory is seen in the relative ease with 
which the changed tone is recognized if the change introduces an 
interval previously not present in the melody. Thus, if we count 
a whole-tone as 1, a major third as 2, an augmented fourth 
(diminished fifth) as 3, and so on, a change from 2—2-1-4 to 
2—-1—2—4 will be less noticeable than a change from 2—1—2-2 to 
2-1-3—1, because in the former case the change is merely an 
interchange of intervals, while in the latter case the change intro- 
duces a new interval, although both represent the same absolute 
amount of pitch change: one whole-tone. The explanation can 
also be made on the basis of pitch-proximity, since the introduc- 


ON THE MELODIC RELATIVITY OF TONES 45 


tion of a large interval into a group of small intervals results in 
a strengthening of tone-groups, the first tones of which become 
correspondingly more accentuated, and hence easier of retention. 

It is inadvisable, in the examples used, to attempt to show the 
other types of emphases such as that of pitch-direction, because 
the longest were only of six tones, which is too short to introduce 
one type of emphasis without involving other types for the same 
tone. Moreover, changes in pitch direction were purposely 
avoided wherever possible in this test. The emphasis which a 
tone receives through a change in pitch direction, for example, 
might be complicated by enforced tone-repetition or pitch-prox- 
imity, which would make an isolated treatment impossible. In 
the examples here given, a particular type of emphasis is suff- 
ciently strong to overcome other types, although here, too, the 
tone is probably not entirely free from emphases other than the 
major ones to which the status of the tone has been attributed. 

For this reason I have selected only those examples in which 
marked deviation from the average makes ambiguity rather 
improbable, but not impossible. This restriction is doubly advis- 
able because no attempt has been made to define the absolute value 
of any type of emphasis, nor has the test been given to a suffi- 
ciently large number of students to make the results certain. 
Then, too, the inevitable harmonic associations operate. The 
agreement of the results in all essentials with the theoretical. 
deduction preceding them, and the absence of any major discrep- 
ancy in the test itself, point to a reasonable degree of reliability. 
The test is not included for a too literal acceptance, since it has 
to be repeated on a much larger scale before its value can be estab- 
lished; it serves, however, as a preliminary check on the general 
laws of melodic relationship. 


46 OTTO ORTMANN 


CONCLUSIONS 


1. The psychological status of any tone in any melody is deter- 
mined by its tonal environment, and by its absolute position in 
the pitch and in the time series. 

2. No one tone of any melody can be changed without thereby 
changing, to a great or small extent, the status of every other tone 
in the melody. 

3. No two tones, in any melody, have the same psychological 
status. 

4, Whenever two or more types of emphasis coincide upon a 
tone, that tone “ stands out”’ from the rest. When the types of 
emphasis do not coincide, or, when one type is in conflict with 
another, that tone is obscured by its environment. 

5. The psychological status of any tone in a melody is not a 
constant. It changes as each new tone of the melody is heard. 

6. The melodic relationship of tones is based upon pitch- 
proximity, with which it varies directly. 

7. Any succession of tones, as used in music, is never reacted 
to purely melodically. Harmonic and rhythmic relationships, 
either expressed or implied, are always present to modify the 
purely melodic effects. 

8. When a test is given as herein described, a melodic memory 
of only two tones may be considered very inferior; one of four 
tones, normal, and one of six tones, superior. 

9. The number of tones in a melody is not in itself a complete 
determinant of the memory-span. Melodies with equal numbers 
of tones may still differ in the ease with which they are retained. 

10. Melodic memory is one element of musical talent, and 
may be sufficiently isolated to permit separate grading. 


ON THE MELODIC RELATIVITY OF TONES 47 


REFERENCES 


. Watt, J. S. The Basis of Music. 
. ORTMANN, O. The Sensorial Basis of Music Appreciation. J. of Compar. 


Psychol. 2:3: 
Lapp, G. Te AND WoopworTH, R. S. Physiological Psychology. 
RayMonn, Gs A ged jt: Essentials of Aesthetics. 
DUNLAP, K. On the Extensity Theory of Pitch. Psychol. Rev., 12, 1905. 
Ewa.p, J. R. Erzeugung von Schallbildern in der Camera Acoustica. 
Arch. f. d. ges. Physiol., 76, 147; 93, 485, 1903. 


. SCHAMBAUGH, G. E. On the Anatomy of the Cochlea. Amer. J. of Anat., 


7, 245, 1907. 


. Harvesty, J. Amer. J. of Anat., 8, 109, 1908. 

. Botse, O. B. Harmony Made Practical, 1900. 

. Prout, E. Harmony, 1903. 

. GoetscHius, P. Material of Musical Composition, 1915. 

. Gortscuius, P. Tone-Relations, 1917. 

. Wagner's Handbuch der Physiologie, III, 11, 481. 

. Bissett, A. D. The Role of Anticipation in Music. 

. PoHLMANN, A. Experimentelle Bettrage zur Lehre vom Gedachtniss. 
. Henri, V. Education de la Mémoire. Année Psychol., 8, 1901. 

. StRuUBE, G. Treatise on Harmony, 1923. 

. Goetscutius, P. Tone Relations, 1917. 

. Goetscuius, P. Elementary Counterpoint. 

. FooreE AND SPALDING. Hamony, 1905. 

. Ricuter, F. Harmony, 1912. 

. OrtTMANN, O. The Fallacy of Harmonic Dualism. Musical Quarterly, 


July, 1924. 


. Epert, E.. anD MEUMANN, E. Grundfragen der Psy. des Gedachtnisses. 


Arch. f. d. ges. Psychol., 4, 1905. 


. ORTMANN, O. Weight Discrimination as a Measure of Technical Skill. 


J. of Compar. Psychol., 3, 1. 


. OrtmMaANN, O. Notes on the Nature of Harmony. Musical Quarterly, July, 


1921; 


. Hermuotrtz, H. On the Sensations of Tones. 


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